A logarithmic function is a function that is the inverse of an exponential function.

The purpose of the logarithm is to tell us about the exponent.

Logarithmic functions are used to explore the properties of exponential functions and are used to solve various exponential equations.

Log base 2 is also known as binary logarithm.

It is denoted as (log2n).

Log base 2 or binary logarithm is the logarithm to the base 2.

It is the inverse function for the power of two functions.

Binary logarithm is the power to which the number 2 must be raised in order to obtain the value of n.

Here’s the general form.

\[{\mathbf{x}}{\text{ }} = {\text{ }}{\mathbf{lo}}{{\mathbf{g}}_{\mathbf{2}}}{\mathbf{n}}\;\;\;\boxed{} - - - - - - - - - \boxed{}{{\mathbf{2}}^{\mathbf{x}}} = {\text{ }}{\mathbf{n}}\]

There are a few properties of logarithm functions with base 2. They are listed in the table below.

Since we are discussing log base2, we will consider the base to be 2 here.

Example 1 –

Log 40, which can be further written as, \[Log{\text{ }}\left( {20 \times {\text{ }}2} \right)\] by-product rule

\[ = {\text{ }}log{\text{ }}20{\text{ }} + {\text{ }}log{\text{ }}2\] which is equal to log 40

Example 2 –

Find the value of log4(4)?

logb(b) = 1, by identity rule

Therefore, log4(4) = 1.

The logarithm can be in the form of log base e or log base 10 or any other bases. Here’s the general formula for change of base -

\[{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{b}}}\;{\mathbf{x}}{\text{ }} = {\text{ }}\frac{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{x}}}}{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{b}}}}\;\]

To find the value of log base 2, we first need to convert it into log base 10 which is also known as a common logarithm.

\[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}x = \frac{{ln\left( x \right)}}{{ln\left( 2 \right)}}\]

Common Logarithmic Function or Common logarithm is the logarithm with base equal to 10.

It is also known as the decimal logarithm because of its base.

The common logarithm of x is denoted as log x.

Example: log 100 = 2 (Since 102= 100).

This is how to find log base 2 -

According to the log rule,

Log Rule -

\[log_{b}(x) = y\]

\[b^{y} = x\]

Suppose we have a question, log216 = x

Using the log rule,

\[\;{2^x} = {\text{ }}16\]

We know that 16 in powers of 2 can be written as \[\left( {2 \times 2 \times 2 \times 2{\text{ }} = 16} \right){\text{ }},{2^x} = {2^4}\]

Therefore, x is equal to 4.

Question 1) Calculate the value of log base 2 of 64.

Solution) Here,

X= 64

Using the formula, \[Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}X = \frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Log base 2 of 64 =\[\frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6\].

Therefore, Log base 2 of 64 = 6

Question 2) Find the value of log2(2).

Solution) To find the value of log2(2) we will use the basic identity rule,

\[lo{g_b}\left( b \right){\text{ }} = {\text{ }}1,\].

Therefore, log2(2) = 1.

Question 3) What is the value of log 2 base 10?

Solution) The value of log 2 base 10 can be calculated by the rule,

\[Lo{g_a}\left( b \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_{10}}\left( 2 \right){\text{ }} = \frac{{\log 2}}{{\log 10}}\; = {\text{ }}0.3010\].

Therefore, the value of log 2 base 10 = 0.3010.

Question 4) What is the value of log 10 base 2?

Solution) The value of log 10 base 2 can be calculated by the rule,

\[Lo{g_b}\left( a \right){\text{ }} = \frac{{\log b}}{{\log a}}\].

\[Lo{g_2}\left( {10} \right){\text{ }} = \frac{{\log 10}}{{\log 2}}\; = {\text{ 3}}{\text{.3 = 2}}\].

Therefore, the value of log 10 base 2 = 3.32.

FAQ (Frequently Asked Questions)

1. What do you mean by log 2?

In mathematics log 2 is equal to log (2, x). It represents the logarithmic value of x with base equal to 2.

2. How to find log base 2 on a calculator?

Calculating logarithm with base 2 is an easy task on a calculator. For example, you want to calculate the logarithm base 2 of 8, then these steps show how to find log base 2–

3. What is the value of log 2 Base 2?

The value of log 2 base 2 is equal to 1.

4. How to calculate log base 2 without a calculator?

According to the log rule, this is how to calculate log base 2.

According to the log rule, this is how to calculate log base 2.

Suppose we have a question, log_{2}16 = x

Using the log rule,

2^{x} = 16

We know that 16 in powers of 2 can be written as (2×2×2×2 =16)

2^{x} =2^{4}

Therefore, x is equal to 4.